Below Absolute Zero—What Does Negative Temperature Mean?

Questions: What is negative temperature? Can you really make a system that has a
temperature below absolute zero? Can you even give any useful meaning to the
expression "negative absolute temperature"?

Answer: Absolutely. :-)

Under certain conditions, a closed system can be described by a negative
temperature, and, surprisingly, be hotter than the same system at any positive
temperature. This article describes how it all works.

Step I: What is "Temperature"?

To get things started, we need a clear definition of "temperature." Actually various
kinds of "temperature" appear in the literature of physics (e.g., kinetic temperature,
color temperature). The relevant one here is the one from thermodynamics, in some
sense the most fundamental.

Our intuitive notion is that two systems in thermal contact should exchange no heat, on
average, if and only if they are at the same temperature. Let's call the two systems
S1 and S2. The combined system, treating S1 and S2 together, can be S3. The
important question, consideration of which will lead us to a useful quantitative
definition of temperature, is "How will the energy of S3 be distributed between S1 and
S2?" I will briefly explain this below, but I recommend that you read K&K,
referenced below, for a careful, simple, and thorough explanation of this important and
fundamental result.

With a total energy E, S has many possible internal states (microstates). The
atoms of S3 can share the total energy in many ways. Let's say there are N different
states. Each state corresponds to a particular division of the total energy in the
two subsystems S1 and S2. Many microstates can correspond to the same division, E1
in S1 and E2 in S2. A simple counting argument tells you that only one particular
division of the energy, will occur with any significant probability. It's the one
with the overwhelmingly largest number of microstates for the total system S3. That
number, N(E1,E2) is just the product of the number of states allowed in each subsystem,
N(E1,E2) = N1(E1)*N2(E2), and, since E1 + E2 = E, N(E1,E2) reaches a maximum when N1*N2 is
stationary with respect to variations of E1 and E2 subject to the total energy constraint.

For convenience, physicists prefer to frame the question in terms of the logarithm of
the number of microstates N, and call this the entropy, S. You can easily see from
the above analysis that two systems are in equilibrium with one another when
(dS/dE)1 = (dS/dE)2, i.e., the rate of change of entropy, S, per
unit change in energy, E, must be the same for both systems. Otherwise, energy will
tend to flow from one subsystem to another as S3 bounces randomly from one microstate to
another, the total energy E3 being constant, as the combined system moves towards a state
of maximal total entropy. We define the temperature, T, by 1/T = dS/dE, so that the
equilibrium condition becomes the very simple T1 = T2.

This statistical mechanical definition of temperature does in fact correspond to your
intuitive notion of temperature for most systems. So long as dS/dE is always
positive, T is always positive. For common situations, like a collection of free
particles, or particles in a harmonic oscillator potential, adding energy always increases
the number of available microstates, increasingly faster with increasing total
energy. So temperature increases with increasing energy, from zero, asymptotically
approaching positive infinity as the energy increases.

Step II: What is "Negative Temperature"?

Not all systems have the property that the entropy increases monotonically with
energy. In some cases, as energy is added to the system, the number of available
microstates, or configurations, actually decreases for some range of energies. For
example, imagine an ideal "spin-system", a set of N atoms with spin 1/2 on a
one-dimensional wire. The atoms are not free to move from their positions on the
wire. The only degree of freedom allowed to them is spin-flip: the spin of a given
atom can point up or down. The total energy of the system, in a magnetic field of
strength B, pointing down, is (N+ - N-)*uB, where u is the magnetic moment of each atom
and N+ and N- are the number of atoms with spin up and down respectively. Notice
that with this definition, E is zero when half of the spins are up and half are
down. It is negative when the majority are down and positive when the majority are
up.

The lowest possible energy state, all the spins pointing down, gives the system a total
energy of -NuB, and temperature of absolute zero. There is only one configuration of
the system at this energy, i.e., all the spins must point down. The entropy is the
log of the number of microstates, so in this case is log(1) = 0. If we now add a
quantum of energy, size uB, to the system, one spin is allowed to flip up. There are
N possibilities, so the entropy is log(N). If we add another quantum of energy,
there are a total of N(N-1)/2 allowable configurations with two spins up. The
entropy is increasing quickly, and the temperature is rising as well.

However, for this system, the entropy does not go on increasing forever. There is
a maximum energy, +NuB, with all spins up. At this maximal energy, there is again
only one microstate, and the entropy is again zero. If we remove one quantum of
energy from the system, we allow one spin down. At this energy there are N available
microstates. The entropy goes on increasing as the energy is lowered. In fact
the maximal entropy occurs for total energy zero, i.e., half of the spins up, half down.

So we have created a system where, as we add more and more energy, temperature starts
off positive, approaches positive infinity as maximum entropy is approached, with half of
all spins up. After that, the temperature becomes negative infinite, coming down in
magnitude toward zero, but always negative, as the energy increases toward maximum.
When the system has negative temperature, it is hotter than when it is has
positive temperature. If you take two copies of the system, one with positive and
one with negative temperature, and put them in thermal contact, heat will flow from the
negative-temperature system into the positive-temperature system.

Step III: What Does This Have to Do With the Real World?

Can this system ever by realized in the real world, or is it just a fantastic invention of
sinister theoretical condensed matter physicists? Atoms always have other degrees of
freedom in addition to spin, usually making the total energy of the system unbounded
upward due to the translational degrees of freedom that the atom has. Thus, only
certain degrees of freedom of a particle can have negative temperature. It makes
sense to define the "spin-temperature" of a collection of atoms, so long as one condition
is met: the coupling between the atomic spins and the other degrees of freedom is
sufficiently weak, and the coupling between atomic spins sufficiently strong, that the
timescale for energy to flow from the spins into other degrees of freedom is very large
compared to the timescale for thermalization of the spins among themselves. Then it
makes sense to talk about the temperature of the spins separately from the temperature of
the atoms as a whole. This condition can easily be met for the case of nuclear spins
in a strong external magnetic field.

Nuclear and electron spin systems can be promoted to negative temperatures by suitable
radio frequency techniques. Various experiments in the calorimetry of negative
temperatures, as well as applications of negative temperature systems as RF amplifiers,
etc., can be found in the articles listed below, and the references therein.